A Short Introduction to Logic Summer School on Proofs as Programs 2002 Eugene (Oregon)

1. A Short Introduction to LogicSummer School on Proofs as Programs2002 Eugene (Oregon) Stefano Berardi Università di Torino stefano@di.unito.it http://www.di.unito.it/~stefano

2. The text of this short course on Logic, together with the text of the next short course onRealizability, may be found in the home page of the author http://www.di.unito.it/~stefano (look for the first line in the topic TEACHING)

3. Plan of the course • Lesson 1.Propositional Calculus. Syntax and Semantic. Proofs (Natural Deduction style). Completeness Result. • Lesson 2. Predicate Calculus. Syntax and Semantic. Proofs (Natural Deduction style). • Lesson 3.Gödel CompletenessTheorem. Validity. Completeness. • Lesson 4. Strong Normalization. Intuitionistic Logic. Strong Normalization. Structure of Normal proofs. • Next Course: Realization Interpretation.

4. Reference Text • Logic and Structure. Dirk van Dalen. 1994, Springer-Verlag. Pages 215.

5. Using the Textbook • What we skipped: • Model theory of Classical Logic (most of §3) • Second Order Logic (§ 4) • Model theory of Intuitionistic Logic (in §5) • Roughly speaking: Lessons 1,2,3,4 correspond to sections §1, §2, §3 and 4, §5 and 6 • of Van Dalen’s textbook. • Roughly speaking (and on the long run): in these Course Notes, one slide corresponds to one page of Van Dalen’s book.

6. Lesson 1 Propositional Calculus Syntax Semantic Proofs Completeness

7. Plan of Lesson 1 • We will quickly go through Syntax and Semantic of Propositional Calculus again. • § 1.1 Syntax. The set of formulas of Propositional Calculus. • § 1.2 Semantic. Truth tables, valuations, and tautologies. • We will really start the course from here: • § 1.3 Proofs. We introduce Natural Deduction formalization of Propositional Calculus. • § 1.4 Completeness. We prove that logical rules prove exactly all “true” propositions. • Forthcoming Lesson: First Order Logic

8. § 1.1 Syntax • The symbol of the language. • Propositional symbols: p0, p1, p2, … • Connectives: (and),  (or),  (not),  (implies),  (is equivalent to),  (false). • Parenthesis: (, ).

9. § 1.1 Syntax • The set PROP of propositions: the smallest closed under application of connectives: • PROP • pi PROP for all iN • PROP  ()PROP • ,PROP  (), (  ), (), () PROP

10. § 1.1 Syntax • Examples: • (p0) • ((p0)) • (p0  (p1  p2)) • (p0 (p1 p2)) • Correct expressions of Propositional Logic are full of unnecessary parenthesis.

11. § 1.1 Syntax • Abbreviations. Let c=, , . We write p0 c p1 c p2 c … • in the place of (p0 c (p1 c (p2 c …))) • Thus, we write p0  p1  p2, p0p1 p2, … • in the place of (p0  (p1  p2)), (p0 (p1 p2))

12. § 1.1 Syntax • We omit parenthesis whenever we may restore them through operator precedence: •  binds more strictly than , , and ,  bind more strictly than , . • Thus, we write: • p0 for ((p0)), • p0  p1 for ((p0 ) p1) • p0  p1 p2 for ((p0 p1)  p2), …

13. § 1.1 Syntax • Outermost symbol. The outermost symbol of , pi,, (), (), (), () • are, respectively: ,pi,, ,,,

14. § 1.1 Syntax • Immediate Subformulas : • Of  and pi are none • Of  is  • Of (), (  ), (), () are ,  •  is a subformula of  iff there is some chain =0, …, n=, each formula being some immediate subformula of the next formula. • Subformulas of =((p0 p1)  p2) are:  itself, (p0 p1), p0, p1, p2.

15. § 1.2 Semantic • Interpreting Propositional constant and connective. • Each proposition pi may be either T (true) or F (false). •  is always F (false). • , , , ,  are interpreted as unary or binary map (or Truth Tables), computing the truth of a statement , (), (), (), (), • given to the truth of immediate subformulas , .

16. § 1.2 Semantic • Truth table of .

17. § 1.2 Semantic • Truth table of .

18. § 1.2 Semantic • Disjunction is taken not exclusive: if, then both ,  may be true.

19. § 1.2 Semantic • Implication is “material”:  is true also for unrelated statements , : it only depends on the truth values of , .

20. § 1.2 Semantic • Equivalence  is identity of truth values.

21. § 1.2 Semantic • Inductive definition. Fix any set I, any map v:NI, any bI, and for any unary (binary) connective c, some unary (binary) map Tc on I. • Then there is exactly one map h:PROPI, such that: • f(pi) = v(i) I for all iN, • f() = b I • f() = T(f()) I • f( c ) = Tc(f(), f()) I for all binary connectives c

22. § 1.2 Semantic • A Valuation is any map v:N{T,F}, assigning truth values to Propositional constants. • Interpreting Propositional formulas. Any valuation v may be extended by an inductive definition to some map h:PROP{T,F}, by: • mapping  into b=False, • using, as Tc, the truth table of connective c= , , , , . • For all  PROP, we denote h() by []v {T,F}

23. § 1.2 Semantic • Let PROP. • Tautologies.  is a tautology iff for all valuations v we have []v =T. • Contradictions.  is a contradiction iff for all valuations v we have []v =F. • Tautology conveys our intuitive idea of being “logically true”, or “true no matter what the Propositional constants are”.

24. § 1.2 Semantic • Some examples of tautologies • Double Negation Law:   . • Excluded Middle:   . • An easy exercise: check that    is a tautology, i.e., that []v = True • for all valuations v:N{T,F}.

25. § 1.3 Proofs • Formal Proofs. We introduce a notion of formal proof of a formula : Natural Deduction. • A formal proof of  is a tree • whose root is labeled , • and whose children are proofs of the assumptions 1, 2, 3, … of the rule r we used to conclude .

26. § 1.3 Proofs • Natural Deduction: Rules. For each logical symbol c=, , , , and each formula with outermost connective c, we give: • A set of Introduction rules for c, describing under which conditions  is true; • A set of Elimination rules for c, describing what we may infer from the truth of . • Elimination rules for c are justified in term of the Introduction rules for c we chose.

27. § 1.3 Proofs • Natural Deduction: the missing connectives. • We treat , , • as abbreviating (), ()(), • We do not add specific rules for the connectives , .

28. § 1.3 Proofs • Natural Deduction: notations for proofs. • Let  be any formula, and  be any unordered (finite or infinite) list of formulas. We use the notation  …  • abbreviated by |- , for: • “there is a proof of  whose assumptions are included in ”.

29. § 1.3 Proofs • Natural Deduction: crossing assumptions. • we use the notation ,  …  • for: “we drop zero or more assumptions equal to  from the proof of ”. \

30. § 1.3 Proofs • Natural Deduction: assumptions of a proof 1 2 3 … r --------------------------------  • are inductively defined as: • all assumptions of proofs of 1, 2, 3, …, minus all assumptions we “crossed”.

31. § 1.3 Proofs • Identity Principle: The simplest proof is:  • having 1 assumption, , and conclusion the same . • We may express it by: |-, for all  • We call this proof “The Identity Principle” (from  we derive ).

32. § 1.3 Proofs • Rules for  • Introduction rules: none ( is always false). • Elimination rules: from the truth of  (a contradiction) we derive everything:  ----  • If |- , then |-, for all 

33. § 1.3 Proofs • Rules for  • Introduction rules:   --------    • If |-  and |-  then |-   

34. § 1.3 Proofs • Elimination rules:     -------- ------- •  • If |-   , then |-  and |- 

35. § 1.3 Proofs • Rules for  • Introduction rules:   -------- -------     • If |-  or |- , then |-   

36. § 1.3 Proofs • Elimination rules:   … …     --------------------------------------  • If |-    and ,|- and , |-, then |- • We may drop any number of assumptions equal to  (to ) from the first (from the second) proof of  \ \

37. § 1.3 Proofs • Rules for  Introduction rule:  …  --------  • If , |- , then |- • We may drop any number of assumptions equal to  from the proof of . \

38. § 1.3 Proofs • Elimination rule:   ----------------  • If |- and |-, then  |- .

39. § 1.3 Proofs • The only axiom not associated to a connective, nor justified by some Introduction rule, is Double Negation:  ….  ---  • If , |- , then |- • We may drop any number of assumptions equal to  from the proof of . \

40. § 1.3 Proofs • Lemma (Weakening and Substitution). • If |- and p, then p|-. • If |- and , |-, then |- . • Proof. • Any proof with all free assumptions in  has all free assumption in p. • Replace, in the proof of  with free assumptions all in ,, all free assumptions  by a proof of  with all free assumptions in .

41. § 1.4 Completeness • Definition (Validity). |- is valid iff for all valuations v such that v(){True}, we have v()=True (iff for no valuation v we have v(){True}, v()=False). • Validity conveys the idea “|- is true no matter what the Propositional constants are”. • Definition (Consistency).  is consistent iff (|-) is false (if  does not prove ). • Definition (Completeness).  is complete iff for all propositions , either |- or |- .

42. § 1.4 Completeness • Correctness. If |- is true in Natural Deduction, then |-is valid. • Proof. Routine. By induction over the proof of |-, considering: • one case for each introduction and elimination rule, • one for the Identity rule, • one for Excluded middle.

43. § 1.4 Completeness • Completeness Theorem. If |- is valid, then then |-is derivable in Natural Deduction. • Proof. • We will pass through many Lemmas: • Lemma 1 (Consistency). If |- is not derivable, then ,  is consistent. • Lemma 2 (Consistent Extension). For all formulas , if  is consistent, then either ,  or ,  is consistent.

44. § 1.4 Completeness • Lemma 3 (Complete Consistent extension). Any consistent set  may be extended to some consistent complete set ’. • Lemma 4 (Valuation Lemma). For every complete consistent consistent set  there is some valuation v such that v()={True}. • Lemma 5 (2nd Valuation Lemma). For every consistent set  there is some valuation v such that v(){True}.

45. § 1.4 Completeness • Lemma 1 (Consistency). If |- is not derivable, then ,  is consistent. • Proof. We will prove the contrapositive: if , |-, then |-. This statement follows by Double Negation.

46. § 1.4 Completeness • Lemma 2 (Consistent Extension). For for all formulas , if  is consistent, then either ,  or ,  is consistent. • Proof. We will prove the contrapositive: if ,|- and ,|-, then |-. • From ,|- and -Intr. we deduce |-. • From |- (by 1 above), the hypothesis ,|-, and Substitution, we conclude |-.

47. § 1.4 Completeness • Lemma 3 (Complete Consistent extension). Any consistent set  may be extended to some consistent complete set ’ (such that for all formulas , either ’|- or ’ |- ). • Proof. Fix any numbering of formulas 0, …, n, … . Let 0, …, n, … be the sequence of sets of formulas defined by: • 0 =  • n+1 = n, n, if n, n is consistent • n+1 = n, n if n, n is not consistent

48. § 1.4 Completeness • Proof of Complete Consistent Extension . • (Consistency) By the Consistent Extension lemma, if n is consistent then n+1 is. Since 0 =  is consistent, then all n are consistent. Thus,  = nn is consistent (a proof of  with assumptions in  would have all assumptions in some n). • (Completeness) By construction,  includes, for all formulas n, either n or n. By the Identity Principle, in the first case |-n, in the second one |-n .

49. § 1.4 Completeness • Lemma 4 (Valuation Lemma). For every complete consistent set  there is some valuation v such that v()={True}. • Proof. Define v()=T iff |-. We have to prove: • v() = F • v(  ) = T  v()=T or v()=T • v(  ) = T  v()=T and v()=T • v(  ) = T  v()=F or v()=T

50. § 1.4 Completeness • v() = F because |- is false, by consistency of .